In [29] L.P. Shilnikov introduced the conditions giving rise to bifurcations of codimension-two termed as orbit-flip and inclination-switch that can only occur in 3D+ systems. Besides that, the inclination-switch bifurcation even in the case of an expanding
saddle with the saddle index satisfying the condition 1/2< ν <1 can also lead to the onset
of stable orbits in the phase space of systems. As such, the occurrence of such a bifurcation is an alarming sign for the Lorenz attractor in the SM-model. Below we will outline the essence of the inclination-switch bifurcation. Its in-depth analysis is given in [1].
Figure 2.7 illustrates the concept of the an inclination-switch bifurcation, which gives
rise to the emergence of a stable orbit. The setup is the following: the 1D separatrix Γ+ of
the saddle of type (2,1) comes back to the saddle along the [vertical] leading direction. We
explore the global map that takes a cross-section, Π, transverse to the stable manifold,Ws,
onto itself along the homoclinic loop. Typically, the local map near the saddle is an expansion for ν <1, i.e. it must stretch a square or a volume. Figure 2.7 sketches how the local map
Figure (2.9) (left) LE-sweep magnification of a Shilnikov flame near the codimension-two point of the [100]-homoclinic loop revealing the fine organization of the bifurcation unfolding and the stability windows. (right) One-parameter cut through the Shilnikov flame (depicted in panel (a)) disclosing cascades of SN and PD bifurcations within it, as well as the occurrence of the secondary, [100.100] and [100.001], homoclinics.
takes a small interval d1 1 on Π into d2 ∼ dν1 > d1. Let us picture an evolution, along
the separatrix loop, of a piece, M, of a leading manifold, being defined locally and tangent
to a span of the eigenvectors corresponding to the leading stable and unstable characteristic
exponents,s2 <0< s1 , resp., of the saddle. As M is dragged away from the saddle by the
outgoing separatrix, it starts curving so that it hits the cross-section, Π, with a transversally
squeezed hook due to the strongly stable exponent, s3 < s2. Because of bending, the image
of d2 becomes shorter than the original, d1, i.e. Td1 < d1 which was not the case prior to
the bifurcation when the overall map was a stretching one. In the aftermath of bending, the global map T becomes a contraction after it overcomes the persistent stretching effect
of the local map near the saddle. This map makes the image TΠ1 of the right (relative to
the stable manifold, Ws, of the saddle) portion, Π, of the cross-section stretch and bend,
so that it looks like a hook or a Smale horseshoe. As such, the map may gain stable fixed points coexisting along with saddle periodic ones.
The 2D return map near the primary homoclinic butterfly of two separatrix loops of a saddle is a core of the geometric model of the Lorenz attractor proposed in [6]. The map is
Figure (2.10) (a){6−20} and (b) {17−20}-kneading ranges revealing a fine structure and self- similarity of the fractal border between the regions of simple dynamics (solid color) and complex chaotic dynamics.
supposed to meet a few analytical conditions guaranteeing that a system in question possesses a genuine chaotic attractor without stable orbits and homoclinic tangencies. A violation of the conditions occurs on a boundary of its existence region. Near the aforementioned codimension-two bifurcations the 2D map can be further reduced to a simplified 1D map (Fig. 2.7) in the following form [1]:
ξn+1 =
µ+A|ξn|ν +o(|ξn|2ν)
·sign(ξn), (2.4)
here 1/2< ν =|λ2|/λ1 <1 is the saddle index,µcontrols the distance between a separatrix,
Γ+, and the stable manifold, W2 of the saddle at the origin, and A is the separatrix value
[40]. The termo(|ξn|2ν) is no longer negligible whenever|A| 1 near the inclination-switch
bifurcations. The top right panels in Fig. 2.7 illustrate the geometry of the map for positive
and negative A. One can figure from the geometry of the hooked map that the unfolding
bifurcations of fixed points, as well as double homoclinics. Say, if the inclination-switch occurs at the homoclinic loop with the [10] kneading, there will be a couple of bifurcation curves of double homoclinics, [10.10] and [10.01] emerging from the codimension-two points.
An alternative, though expensive, solution for locating the curve A = 0 in the parameter
space is by detecting the hooks in the return map generated by successive minima of the
z-variable. Two such maps above and below the curve A = 0 at two locations, α = 0.39,
and λ = 0.79 and λ = 0.77, are presented in the bottom right panel of Fig. 2.7. The later
map features a second smooth critical point in addition to the cusp that will break down the instability and lead to the occurrence of stable periodic orbits in the phase space and
stability windows in the parameter space of the SM-model. We note that at lower λ and α
values there are other curves similar to A = 0 [35]. Crossing down each such curve makes
the return map bend again to gain additional turns. With every new turn, the map near the
saddle starts appearing like the Poincar´e map near a Shilnikov saddle-focus. The distinction
though is that the spiraling saddle-focus map generates countably many Smale horseshoes, whereas the map near such twisting saddle has only a finite number of turns.